Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra
Genqiang Liu, Mingjie Li
TL;DR
The work establishes a concrete bridge between irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules and finite-dimensional modules over the minimal nilpotent $W$-algebra $W(e)$. By constructing an injective Miura-type map $\tau: W\to U(\mathfrak{gl}_n)$ and analyzing finite-dimensional $W$-modules through the auxiliary algebra $\sigma\tau(W)$, the authors show that every finite-dimensional irreducible $W$-module is a quotient of some finite-dimensional $\mathfrak{gl}_n$-module via $V(\lambda)^\tau$. They then realize all irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules explicitly as irreducible quotients of induced modules $G_\mu(V(\lambda)^\tau)$, with precise nonintegrality conditions on parameters, and connect these realizations to Shen–Larsson modules and twisted parabolic Verma modules. The results provide a transparent, algebraically explicit route to the classification and construction of irreducible cuspidal modules, avoiding earlier techniques like twisted localization and coherent families.
Abstract
A weight $\mathfrak{sl}_{n+1}$-module with finite-dimensional weight spaces is called a cuspidal module, if every root vector of $\mathfrak{sl}_{n+1}$ acts injectively on it. In \cite{LL}, it has been shown that any block with a generalized central character of the cuspidal $\mathfrak{sl}_{n+1}$-module category is equivalent to a block of the category of finite-dimensional modules over the minimal nilpotent finite $W$-algebra $W(e)$ for $\mathfrak{sl}_{n+1}$. In this paper, using a centralizer realization of $W(e)$ and an explicit embedding $W(e)\rightarrow U(\mathfrak{gl}_n)$, we show that every finite-dimensional irreducible $W(e)$-module is isomorphic to an irreducible $W(e)$-quotient module of some finite-dimensional irreducible $\mathfrak{gl}_n$-module. As an application, we can give very explicit realizations of all irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules using finite-dimensional irreducible $\mathfrak{gl}_n$-modules, avoiding using the twisted localization method and the coherent family introduced in \cite{M}.